Researching the change of the operating frequencies in the case of inequality of the output voltages of the comparator within the structure of an integrating measuring strain gauge converter

One of the major errors directly influencing the metrological characteristics of the integrating measuring strain gauge converter is the inequality of the output voltages of the comparator. The current paper explores the effect of the voltages variation at the output of the comparator in the case of a bipolar power supply of the converter. The output data is obtained by modeling the equation of conversion in the MATLAB environment. The fore-mentioned problem is investigated assuming up to 20% inequality of the output voltages compared to the supply voltage and a bilateral change of the load on the strain gauges. A regression analysis is performed checking the suitability of a linear, quadratic and cubic model. It shows that the coefficient of determination is highest for the cubic model and relevant conclusions are made.

Strain gauges measuring converters are intended to amplify and convert the small output voltages that are obtained in disbalance of strain gauges bridge, which is known as "Wheatstone bridge". Every "Wheatstone bridge" consists of two or four strain gauges that are connected in half bridge or full bridge. The output voltage is proportional to the deformation of the strain gauges sensors. Depending on the mode of guidance (method of orientation) for bonding, the strain gauges can measure forces, moments, weight, etc.
The transformation of the change of the strain gauges deformation into frequency deviation is done by integrating converters working on the method of ramp right conversion. The method itself is well known and reported by many scientists: (Шахов, 1986) , (Mochizuki,1996), (Kaliyugavaradan, 2000), (Madhu, 2009), (Гигов, 2013), (Станков, 2014) and many others. Essential advantages of the method are simplicity circuits and high linearity conversion. The main disadvantages are the requirements for using high performance elements, fast operational amplifiers, and accurate selection of measurement ranges. Therefore, there is no coincidence that most schemes of integrating converters have patent protection rights, namely (Mильченко, Романов, 1980), (Гигов, Гутников, 1981), (Гигов, Янков, 1986), (Glimert 1994), (Василев, Громков, 2009) and many others.
Here is investigated a converter of the disbalance of the resistance of a strain gauge bridge into frequency deviation; its block scheme is given in Figure 1 as well (Stoyanov, 2014).
The converter in Fig. 1 comprises as follows: integrator 4, comparator 1, strain-gauge resistive bridge 2, a differential amplifier 3 and a voltage divider 5 and 6. The power diagonal of the bridge is connected between the output of the comparator and the common ground. Measuring is done diagonally to the inputs of the differential amplifier DA, whose output is connected to the inverting input of the integrator I. The output of the integrator is connected to the inverting input of the comparator C, whose noninverting input and output are associated with the output of the converter F.

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The output frequency is determined by the following equation: where: -T -the output period of the converter; -k DA -the gain of the instrumental amplifier; - I -time constant of the integrator; -δR -relative change of the resistance of the strain gauges load (deformation).
The proposed converter has very good metrological characteristics in bilateral change to the load. A prototype has been developed, and a patent application has been made (№111382, made on 25.01.2012). (Гигов, Станков, Стоянов, 2014).
On the basis of the main metrological characteristics of the converter, which have been received up to this moment, there is need of further analysis of the errors that determine the appearance of nonlinearity and that change the output frequency (Гигов, 2013).
One of the main errors that influence directly the metrological characteristics is the inequality of the output voltage of the comparator. This leads to adding or subtracting the value of the error of the output voltage to the values of the positive or negative output voltages and [Гигов, 2013].
The values of the output voltages of the comparator  out U and  out U are as follows: (3) For the corrected period T * the following is obtained: (4) After substitution with 2 and 3 in 4 the following is obtained: Here the following designation is introduced: After substitution of (6) and (5) in (1) and the conversion, the corrected output frequency f * of the equation of the conversion is obtained (7): where δU out is the relative error of the output voltage

ISSN 2603-316X (Online)
Published: 2017-12-28 Table 1 shows the data from the change of the output frequency of the comparator in modeling the equation of conversion and the influence of error in MATLAB environment (Stoyanov, 2014). A chart is built too, and it shows the time of changing of the values under load R in 0,1 Ω and voltage variation U in the interval from 0 to 1V. At the output voltage of the converter 5 V, the change of 1V represents 20% of the change of the output voltage. This amendment is sufficient to research the behavior of the converter at the specified input parameters. Table 1. Modification of the output frequency f depending on the amendment to U and R The chart in Figure 2 shows the dependencies of the change of the output frequency on the difference of the output voltages of the comparator at a fixed disbalance. The conducted modeling shows that the created simulation model of the measurement converter is highly dependent on the inequality of the absolute values of the output voltages of the comparator. The received non-linearity influences strongly the metrological characteristics of the converter.
A more precise estimate of the impact of the changing tensions could be made with the aid of mathematical statistics.
It is necessary to establish the relationship with a change of the voltage difference when working with a certain load (disbalance of the system) of the change of the output frequency. In the converter all of the control factors are quantitative, and the links between them are described and analyzed mathematically by the methods of regression analysis (Митков, 2011).

Main part
The object of research is presented in Fig.1. In this case there is a manageable factor x, one output parameter Y and disturbing influence . Since the value of the parameter f is formed both by U and Where: Y=f x=U  -aggregate disturbing influence, which is caused by the uncontrollable factors w 1 ,w 2 ,w i (x) -function of the factor x The theoretical model of the researched converter will have the following form: (9) at R=const. The general appearance of the regression model is selected on the base of geometric submitted experimental data. Searched regression model can be written as follows: Where  0 ; 1 ; 2 ; i are the regression coefficients and =(x) It is necessary that the function f(x) be approximated to one or several functions in order to determine the most reliable regression model: Because of the random error  in the experimental data and because of the final number of such data, one does not receive the exact values of the parameters  0 ; 1 ; 2 ; I but their ratings b 0 ; b 1 ; b 2 ; b i , which are defined as experimental regression coefficients. Thereby the experimental model is obtained on the basis of the theoretical model.
Where i = 1, 2, 3 ... .N is the number of points of the experiment In the deduced formulas of the metrological analysis one can see that the influence of the change of the tension is of the second degree. Taking into account dependencies of mathematical modeling and type of the chart, it is expected that the presentation of the regression model will be a second order polynomial, nonlinear internal to the factor U.
By means of statistical analysis it is necessary to determine the degree of definiteness of the researched model, to determine the portion of change of the parameter Y, which describes the model, the determine the adequacy of the regression models, to determine the significance of the regression coefficients b 0 b 1 ,b 2 , to perform analysis of the residues  j =y j -, and then to specify the final form of the model on the base of a comparative analysis of the three proposed models.  In Table 2 are given the calculated coefficients of determination R and R 2 for the three models. The value of the coefficient indicates the percentage of change in the frequency of Y from tension factor and the remainder to 100% due to other unmanageable factors. Generally, the closer R 2 is to 1, the better is the selected model`s description of the change of the output value of the investigated factors (Митков, 2011). Variations of the researched variable (Adjusted R 2 ), as well as the standard error in the calculations (Stad. Error of the Estimate), are very small, which confirms the right choice of regression model.
In Table 2 it is seen that the coefficients of determination are weakly influenced by the change in resistance, and at values 0Ω and 0,5Ω are identical. This suggests regression analysis to be performed only maximum and minimum value of change of resistance (load) of the sensors of the converter.
In Figure 3 is shown a regression model at maximum negative load; in Figure 4 it is shown under zero load; and in Figure   Evaluation adequacy of the regression models is done using the criterion of Fisher, which is the ratio of residual dispersion in relation to the dispersion of the reproducibility for each model-table 3. The greater the value of the criterion of Fisher, the more adequate the model is [10]. It is necessary, however, to compare initially the calculated value with the critical value of the criterion of Fisher -F kr at significance level  = 0,05. If the value is lower than critical, the model is not adequate. Table 4 gives the values that characterize the significance of the coefficients (parameters) of the model b 0 , b 1 , b 2 b 3 and the regression coefficients  0 ,  1 ,  2,  3 .
The calculated value of the criterion of Student, for t i -in these cases where N on number of experiments -is compared with data references particular for the degrees of freedom k = N-1 and γ = 1-α, while α = 0,05 -level of significance (Митков, 2011). A verification is made that t i > t kr . If this inequality is satisfied, that means that the respective coefficient of linear, quadratic or cubic model is significant.  The developed model of measurement converter is highly dependent on the inequality of the absolute values of the output voltages of the comparator.
 The coefficient of determination is the highest for the cube model, i.e. it most accurately describes the changes in frequency under the influence of confounding factors.  Assessment evaluation of the adequacy of regression models is performed using the criterion of Fisher. Fisher's criterion is highest for cube model and does not depend on the load value on the strain gauges.
 The coefficient of determination and values of Fisher's criterion are identical across the whole measuring range. Thus, the high linearity of the converter in the research range is confirmed.
 Student's criterion does not run for negative values of t i . All positive values t i are significant. This results in a further simplification of regression models and a proof for the significance and relevance of the created models.
 The obtained results are the basis for further improvement of the converter and guaranty of the linearity in the whole measurement range.

Conclusion
The usage of the regression analysis turns out to be an appropriate decision for testing converter work in different modes. The proven capability to cube model suggests an extremely precise execution of power and measuring circuits.
It is important and necessary condition for the normal operation of the converter is to avoid differences in the output of the comparator tensions, especially due to their strong influence in the equation of transformation and their combined disturbance influence caused by unmanageable factors w 1 , w 2 , w i .